Theorem suplocexprlemlub 8085

Description: Lemma for suplocexpr 8086. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemlub	⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑥,𝐴,𝑦   𝑧,𝐵   𝜑,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑤,𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦<P 𝐵)
2		ltrelpr 7866	. . . . . . . 8 ⊢ <P ⊆ (P × P)
3	2	brel 4825	. . . . . . 7 ⊢ (𝑦<P 𝐵 → (𝑦 ∈ P ∧ 𝐵 ∈ P))
4	3	simpld 112	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝑦 ∈ P)
5	4	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦 ∈ P)
6	3	simprd 114	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝐵 ∈ P)
7	6	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝐵 ∈ P)
8		ltdfpr 7867	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝐵 ∈ P) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
9	5, 7, 8	syl2anc 415	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
10	1, 9	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))
11		simprrr 546	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (1st ‘𝐵))
12		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
13	12	fveq2i 5696	. . . . . . . . 9 ⊢ (1st ‘𝐵) = (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
14		npex 7834	. . . . . . . . . . . . 13 ⊢ P ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → P ∈ V)
16		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
17		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
18		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
19	16, 17, 18	suplocexprlemss 8076	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
20	15, 19	ssexd 4271	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
21		fo1st 6385	. . . . . . . . . . . . 13 ⊢ 1st :V–onto→V
22		fofun 5614	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
24		funimaexg 5463	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V)
25	23, 24	mpan 428	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (1st “ 𝐴) ∈ V)
26		uniexg 4583	. . . . . . . . . . 11 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V)
27	20, 25, 26	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (1st “ 𝐴) ∈ V)
28		nqex 7724	. . . . . . . . . . 11 ⊢ Q ∈ V
29	28	rabex 4278	. . . . . . . . . 10 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V
30		op1stg 6378	. . . . . . . . . 10 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
31	27, 29, 30	sylancl 417	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
32	13, 31	eqtrid 2283	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐵) = ∪ (1st “ 𝐴))
33	32	eleq2d 2308	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
34	33	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
35	11, 34	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ ∪ (1st “ 𝐴))
36		suplocexprlemell 8074	. . . . 5 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
37	35, 36	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
38		simprl 535	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ Q)
39	38	ad2antrr 492	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ Q)
40		simprrl 545	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (2nd ‘𝑦))
41	40	ad2antrr 492	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (2nd ‘𝑦))
42		simpr 110	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (1st ‘𝑧))
43		rspe 2599	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
44	39, 41, 42, 43	syl12anc 1276	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
45	4	ad4antlr 499	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦 ∈ P)
46	19	ad4antr 498	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝐴 ⊆ P)
47		simplr 533	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ 𝐴)
48	46, 47	sseldd 3249	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ P)
49		ltdfpr 7867	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
50	45, 48, 49	syl2anc 415	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
51	44, 50	mpbird 167	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦<P 𝑧)
52	51	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) → (𝑠 ∈ (1st ‘𝑧) → 𝑦<P 𝑧))
53	52	reximdva 2652	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
54	37, 53	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
55	10, 54	rexlimddv 2673	. 2 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
56	55	ex 115	1 ⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ⊆ wss 3220  ⟨cop 3711  ∪ cuni 3933  ∩ cint 3968   class class class wbr 4128   “ cima 4775  Fun wfun 5369  –onto→wfo 5373  ‘cfv 5375  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   <_Q cltq 7646  Pcnp 7652  <_P cltp 7656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6368  df-qs 6807  df-ni 7665  df-nqqs 7709  df-inp 7827  df-iltp 7831

This theorem is referenced by:  suplocexpr  8086